References

CheungHawk01B

N. Cheung, Hawk01-B, version 1, unpublished manuscript, n.d. Supplied excerpts: “Controlled island comparison without a reconstruction overclaim” and “Conclusions and mathematical release,” including eq:Petz and thm:release. Version-qualified documentary citation; no independent verification of the full manuscript is asserted.

CheungHawk02A

N. Cheung, Hawk02-A, version 1, unpublished manuscript, n.d. Supplied excerpt: “Decision ledger and exact ending.”

CheungHawk02C

N. Cheung, Hawk02-C, version 2, unpublished manuscript, n.d. Supplied abstract, gravitational-obligation ledger, and “Discarded probability is not a generic coherent error.”

CheungHawk02D

N. Cheung, Hawk02-D, version 1, supplied manuscript text, n.d. Autonomous quadratic fermionic benchmark; supplied abstract and mathematical release. Author attribution follows the supplied source record.

CheungHawk03A

N. Cheung, Hawk03-A, version 1, unpublished manuscript, n.d. Supplied abstract and “Source-boundary audit and release.”

CheungEn02B

N. Cheung, En02-B, version 1, unpublished manuscript, n.d. Supplied abstract and finite programme-certificate excerpts. Batch and streaming decoder statements refer to its declared perfect-code model.

CheungEn02C

N. Cheung, En02-C, version 1, unpublished manuscript, n.d. Supplied excerpts: “Information availability versus executable recovery” and thm:C1, sequential reference-preserving soundness.

CheungEn02E

N. Cheung, En02-E, version 2, unpublished manuscript, n.d. Supplied abstract and cutoff/interface excerpts. Selected Gaussian-state cutoff identities retain their stated commutation and preparation hypotheses.

CheungEm03C

N. Cheung, Em03-C, version 1, unpublished manuscript, n.d. Supplied excerpts: “Source repairs and logical checks” and “One controlled correction and microscopic dynamics.”

CheungEm03D

N. Cheung, Em03-D, version 1, unpublished manuscript, n.d. Supplied excerpts: “Repairs to the proposed proof architecture” and “Access and effort.”

CheungEm03F

N. Cheung, Em03-F, version 1, unpublished manuscript, n.d. Supplied logical-repair and exterior scalar-source release excerpts. Effective exterior source model; no gravitational identification is imported.

CheungEm03G

N. Cheung, Em03-G, version 1, unpublished manuscript, n.d. Supplied resource ledger and ordered exterior-source release. Its packet decoder is not a coherent local receiver for the present model.

CheungEm03H

N. Cheung, Em03-H, version 8, unpublished manuscript, n.d. Supplied thm:channels, “Complete finite channel interface.”

Watrous

J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.

KSW

D. Kretschmann, D. Schlingemann, and R. F. Werner, “The information-disturbance tradeoff and the continuity of Stinespring's representation,” IEEE Transactions on Information Theory 54 (2008), 1708–1717. doi: 10.1109/TIT.2008.917696. arXiv:quant-ph/0605009.

HaslerHerbst

D. Hasler and I. Herbst, “Ground states in the spin boson model,” arXiv:1003.5923. Used as context for standard field-operator and spin–boson Hamiltonian constructions; no ground-state or endpoint theorem from this reference is assumed for the present many-level source.

FeynmanVernon

R. P. Feynman and F. L. Vernon, Jr., “The theory of a general quantum system interacting with a linear dissipative system,” Annals of Physics 24 (1963), 118–173. doi: 10.1016/0003-4916(63)90068-X.

KourkoulouMaldacena

I. Kourkoulou and J. Maldacena, “Pure states in the SYK model and nearly-${\mathrm{AdS}}_2$ gravity,” arXiv:1707.02325.

MaldacenaStanford

J. Maldacena and D. Stanford, “Comments on the Sachdev–Ye–Kitaev model,” Physical Review D 94 (2016), 106002. doi: 10.1103/PhysRevD.94.106002. arXiv:1604.07818.

ChenQiZhang

Y. Chen, X.-L. Qi, and P. Zhang, “Replica wormhole and information retrieval in the SYK model coupled to Majorana chains,” Journal of High Energy Physics 06 (2020), 121. doi: 10.1007/JHEP06(2020)121. arXiv:2003.13147.

GaikwadEtAl

A. Gaikwad, A. Kaushal, G. Mandal, and S. R. Wadia, “A microscopic model of black hole evaporation in two dimensions,” Journal of High Energy Physics 08 (2023), 171. doi: 10.1007/JHEP08(2023)171. arXiv:2210.15579.

NakayamaMiyataUgajin

Y. Nakayama, A. Miyata, and T. Ugajin, “The Petz (lite) recovery map for the scrambling channel,” Progress of Theoretical and Experimental Physics 2023 (2023), 123B04. doi: 10.1093/ptep/ptad147. arXiv:2310.18991.

PSSY

G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, “Replica wormholes and the black hole interior,” arXiv:1911.11977.

AEMM

A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” Journal of High Energy Physics 12 (2019), 063. doi: 10.1007/JHEP12(2019)063. arXiv:1905.08762.

Mathur

S. D. Mathur, “The information paradox: A pedagogical introduction,” arXiv:0909.1038.

AMPS

A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, “Black holes: Complementarity or firewalls?” arXiv:1207.3123.

SuppliedCatalogue

OpenAI, External mathematics catalogue, supplied manuscript records and excerpts, including families 227, 260, 262, 264, 265, 272, and 273. The supplied record identifies a catalogue dated 7 October 2026. Documentary citation only: the catalogue's claimed proofs are not independently verified here and none is a premise of the microscopic, recovery, or gravitational statements in this paper.